Please note that ambiguous notation will cause your question to be interpretted differently to what you intend.

I did NOT mean -(5!). Just like -5^2 equals -25 WITHOUT parens.
I can rephrase if you like: Why does -5^2=-25 without parens, but -5!
needs parens to get -120? Or if you prefer: Why is -5^2 NOT ambiguous, but -5! is? (I'm don't believe -5! really is ambiguous by the way)

I did NOT mean -(5!). Just like -5^2 equals -25 WITHOUT parens.
I can rephrase if you like: Why does -5^2=-25 without parens, but -5!
needs parens to get -120? Or if you prefer: Why is -5^2 NOT ambiguous, but -5! is? (I'm don't believe -5! really is ambiguous by the way)

I did NOT mean -(5!). Just like -5^2 equals -25 WITHOUT parens.
I can rephrase if you like: Why does -5^2=-25 without parens, but -5!
needs parens to get -120? Or if you prefer: Why is -5^2 NOT ambiguous, but -5! is? (I'm don't believe -5! really is ambiguous by the way)

Gene Klein

-5 is a number. You will find that -5! is interpretted as the factorial of -5, NOT the negative of the factorial of 5.

Your question is one of interpretation. Difficulties of interprettation can be entirely avoided simply by using brackets. But for some unfathomable reason, students just don't want to do this. In fact, students get quite stroppy when told to do this. And even stroppier when their work is misunderstood. And often it is themselves that misunderstand their own work ....

-5 is a number. You will find that -5! is interpretted as the factorial of -5, NOT the negative of the factorial of 5.

Your question is one of interpretation. Difficulties of interprettation can be entirely avoided simply by using brackets. But for some unfathomable reason, students just don't want to do this. In fact, students get quite stroppy when told to do this. And even stroppier when their work is misunderstood. And often it is themselves that misunderstand their own work ....

Just use brackets!

What I cannot fathom is why -5^2 does not need brackets, but (according to you) -5! does. Is that a question that you can answer?
Gene Klein

I think there is not a rule for something like this but the important point is to know that the -5 to the power two equal 25 and the factorial for negative is undefined or ambiguous and the other important thing is to write for example -5 to the power two in a way that the reader can take it for example

if I want to answer it I will consider it like this

maybe another one consider it like this

so if you use brackets you will remove the misunderstanding for the problem

I think there is not a rule for something like this but the important point is to know that the -5 to the power two equal 25 and the factorial for negative is undefined or ambiguous and the other important thing is to write for example -5 to the power two in a way that the reader can take it for example

if I want to answer it I will consider it like this

maybe another one consider it like this

so if you use brackets you will remove the misunderstanding for the problem

so it is important to use brackets

One DOESN'T need brackets for -5^2. There is a rule for it. The answer is -25. My question is, why does (or so it seems) -5! need brackets in order to be clear?
Gene Klein

It depends very much on the context. If you were to write something like 6! – 5! = 720 – 120, it would be clear that the minus is a binary operation, and nobody would see any need for parentheses around the 5! Similarly, in an expression like –5^2, there is a universal convention (apparently known in the USA as PEMDAS) which says that you must evaluate the power before taking the negative. But the expression –5! on its own is unusual, and looks strange when not in some context. The PEMDAS order of priorities does not cover factorials, so any sensible mathematician would follow Mr F's excellent advice and Just use brackets! in order to make the meaning unambiguous.

It depends very much on the context. If you were to write something like 6! – 5! = 720 – 120, it would be clear that the minus is a binary operation, and nobody would see any need for parentheses around the 5! Similarly, in an expression like –5^2, there is a universal convention (apparently known in the USA as PEMDAS) which says that you must evaluate the power before taking the negative. But the expression –5! on its own is unusual, and looks strange when not in some context. The PEMDAS order of priorities does not cover factorials, so any sensible mathematician would follow Mr F's excellent advice and Just use brackets! in order to make the meaning unambiguous.

The convention which you quote includes the factorial. (Although obviously not in the name). The precedence of factorial is equal to exponetiation (Full Disclosure: Some argue that it is greater than exponentiation). -5! may look unusual, but quite frankly, so does
-5^2=-25 (at least to me!). Looking unusual is not the criteria for deciding if () are needed. I know many (non math students) people who would find 3+4x2=11 VERY UNUSUAL. Their surprise at the answer of 11 does not obligate me to include parens.

My question is simple. Just like from a strict order of operations point of view, 3+4x2=11 WITHOUT parens, I want to what is -5! Don't put in parens. Just tell me what it equals.